Group Representations in Two- and Three-Dimensional Kakeya Geometry

Kakeya geometry asks how small a subset of Euclidean space can be while containing a unit line segment in every direction. The problem is central to harmonic analysis because its tube formulations are closely related to maximal operators, Fourier restriction, and oscillatory integrals. This paper provides an analysis of the two-dimensional and three-dimensional Kakeya problems from the perspectives of group action and representation theory. This paper first passes from Kakeya sets to finite families of thin tubes, where dimension is studied through tube number, multiplicity, and union volume. It then describes an oriented line by its direction and transverse position and identify the line space as the homogeneous space L n + ≅ S E ( n ) / ( R × S O ( n − 1 ) ) . The Euclidean action on points and lines induces quasi-regular representations, and the X-ray transform becomes an intertwining operator satisfying X π ( g ) = π ^ ( g ) X . In two dimensions, this framework organizes translation invariance, Fourier analysis, and angular almost orthogonality. In three dimensions, spherical harmonics and affine rescaling organize directional and scale information, while additional positional and multiscale correlations explain why symmetry alone is insufficient. The resulting formulation does not provide a new Kakeya estimate; instead, it gives a unified language for comparing the planar and spatial problems and for isolating the geometric difficulties that remain in higher dimensions.